Dynamic lot-sizing in sequential online retail auctions

Abstract

Retailers often conduct non-overlapping sequential online auctions as a revenue generation and inventory clearing tool. We build a stochastic dynamic programming model for the seller’s lot-size decision problem in these auctions. The model incorporates a random number of participating bidders in each auction, allows for any bid distribution, and is not restricted to any specific price-determination mechanism. Using stochastic monotonicity/stochastic concavity and supermodularity arguments, we present a complete structural characterization of optimal lot-sizing policies under a second order condition on the single-auction expected revenue function. We show that a monotone staircase with unit jumps policy is optimal and provide a simple inequality to determine the locations of these staircase jumps. Our analytical examples demonstrate that the second order condition is met in common online auction mechanisms. We also present numerical experiments and sensitivity analyses using real online auction data.

Introduction

Online auctions of retail goods have become a significant component of modern internet commerce. Several large retailers such as Dell (www.dellauction.com) and Sam’s Club (auctions.samsclub.com) increasingly use online auctions as a revenue generation mechanism (Bapna et al., 2008, Pinker et al., 2010). In combination with scrapping excess inventories to firms like Overstock (www.overstock.com), large retailers also use online auctions as an inventory clearing tool. The auction giant eBay (www.ebay.com) and other similar firms such as Ubid (www.ubid.com) provide auction hosting services to retailers like IBM, Sharp and Fujitsu, and also to individual sellers. Companies like Truition (www.truition.com) and ChannelAdvisor (www.channeladvisor.com) specialize in helping businesses conduct online auctions (Odegaard and Puterman, 2006). Based on empirical data available in Vakrat and Seidmann, 2000, Pinker et al., 2010 have noted that most retail auction websites conduct a sequence of multi-unit auctions of identical items. These auctions were also observed to be the operational norm by Bapna et al., 2008, Pinker et al., 2003, Tripathi et al., 2009. Pinker et al. (2003) have summarized various research issues in such auctions.

Lot-sizes, that is, the number of units to be auctioned in each auction, are one of the key decision variables in sequential auctions (Pinker et al., 2003, Pinker et al., 2010, Segev et al., 2001, Tripathi et al., 2009, Vakrat and Seidmann, 2000). A small lot-size induces bidder competition thus increasing the clearing-price. The total revenue may still be lower than one would hope because the number of units sold is small. Uncertainty in the number of participating bidders (demand) in each auction and that in their bids increases decision complexity. For instance, an auction with too large a lot-size may fail due to insufficient demand. Inventory holding costs and the possibility of scrapping inventory to save and recover some of these costs introduce additional economic tradeoffs.

Two papers have investigated inventory scrapping and/or lot-sizing decisions in sequential online retail auctions (Pinker et al., 2010, Tripathi et al., 2009).

Pinker et al. (2010) studied these problems under the following restrictions: a fixed number of participating bidders in each auction, uniform bid distributions with support [0, 1], and a truth revealing multi-unit Vickrey mechanism. These assumptions enabled them to formulate a deterministic dynamic program, wherein a closed-form lot-sizing policy was derived by equating derivatives of value functions to zero within a backward induction procedure. The optimal lot-size decreased at a constant rate from one auction to the next. This rate increased with inventory holding costs and decreased with the number of bidders per auction. In their model, it was optimal to scrap inventory only one time before beginning the entire sequence of auctions.

Tripathi et al. (2009) also assumed a fixed number of participating bidders in each auction, and employed a multi-unit Dutch mechanism. Using uniform bid distributions, they first optimized the lot-size over a sequence of auctions assuming that the lot-size did not change over time. This led to a simple closed-form lot-size expression that resembled the well-known Economic Order Quantity (EOQ) formula in inventory management (Heyman and Sobel, 2003). They also devised a goal programming method to estimate bid distributions from online bid data.

Segev et al. (2001) focused on predicting auction clearing-prices using an orbit queue Markov chain model, and compared these predictions with data obtained from Onsale (www.onsale.com), a Sillicon Valley start-up company. They proposed a deterministic dynamic programming model for lot-size optimization under the restrictive assumption that all items on sale will be sold owing to a sufficiently large number of participating bidders but did not attempt to solve it.

Odegaard and Puterman (2006) considered an auctioneer with two identical items on hand, and determined an optimal time-point at which the second item should be “released” for an auction. They derived conditions to ensure an optimal control-limit release-time policy. This control-limit was decreasing in holding cost.

Vulcano et al. (2002) studied a problem motivated by airline ticket selling websites like Priceline (www.priceline.com). The seller first observed bids from potential travelers, and then chose how many and which bids to accept, as opposed to publicly pre-committing lot-sizes at the beginning of each auction before receiving bids as practiced in retail auctions (Odegaard and Puterman, 2006, Pinker et al., 2010, Segev et al., 2001, Tripathi et al., 2009). Consequently, they solved a variable supply allocation problem rather than a lot-size optimization problem to obtain a structural result similar to ours but utilized different mathematical analysis and sufficient conditions as developed by Myerson, 1981, Maskin and Riley, 2000. This work was later extended to an infinite-horizon joint auctioning and pricing problem under holding and ordering costs (van Ryzin and Vulcano, 2004).

The basic setting in our paper is similar to Pinker et al., 2010, Tripathi et al., 2009 in that we consider a seller who conducts a sequence of non-overlapping online auctions of retail goods. However, in contrast to their work, we incorporate uncertainty in the number of participating bidders (stochastic demand) in each auction; do not restrict our formulation to any specific clearing-price determination rule; and allow for any bid distribution (see Section 2 for details). To the best of our knowledge, this is the first paper that successfully overcomes all mathematical difficulties introduced by this generalization in the retail pre-committing setting to provide a complete structural characterization of optimal inventory scrapping and lot-sizing policies as in Theorem 2.1.

More specifically, under the second order condition (7) on the single-auction expected revenue function, we show that a threshold inventory-scrapping policy, and a monotone staircase with unit jumps lot-sizing policy are optimal. This condition roughly requires that the marginal single-auction expected revenue, normalized by the probability of sufficient number of bidders participating, be decreasing in lot-size. It is then optimal to scrap all inventory above a time-dependent threshold inventory level, and not to scrap any inventory below the threshold. This threshold equals the inventory level at which the scrap-value of a unit exceeds its marginal value over all remaining auctions. Moreover, if lot-size x is optimal in post-scrapping inventory i, then either lot-size x or lot-size x + 1 is optimal in post-scrapping inventory i + 1. This unit jump in optimal lot-size occurs when the normalized marginal single-auction expected revenue exceeds the discounted marginal value of saving the additional unit for future auctions. See Theorem 2.1 and its proof in Section 3 for precise detailed versions of these statements. Section 3.1 includes several examples where our second order condition is met. Numerical results and sensitivity analyses conducted using real online auction data are presented in Section 4. Limitations and potential extensions of our model are discussed in Section 5.